A Variational Framework for the Algorithmic Complexity of PDE Solutions
Juan Esteban Suarez Cardona, Holger Boche, Gitta Kutyniok
TL;DR
The paper introduces a least-squares variational framework to study the computability and computational complexity of PDE solutions by analyzing discrete gradient flows of variational losses. It connects PDE operator structure (coercivity, ellipticity, convexity) and data regularity to whether solutions are computable in polynomial time, or suffer complexity blowup, via an error-decomposition analysis in finite-dimensional surrogate spaces. The framework yields sufficient conditions for $H^k$-computability, demonstrates polynomial-time solvability for Poisson-type problems under analytic data, and shows blowup scenarios for nonlinear equations like the Eikonal equation, illustrating how regularity loss drives super-polynomial complexity. Together with Sobolev cubatures and computability theory in Banach spaces, the work provides a constructive theoretical bridge between computability, complexity, and variational PDE methods, with practical implications for algorithm design and understanding fundamental limits of numerical PDE solvers.
Abstract
Partial Differential Equations (PDEs) are fundamental tools for modeling physical phenomena, yet most PDEs of practical interest cannot be solved analytically and require numerical approximations. The feasibility of such numerical methods, however, is ultimately constrained by the limitations of existing computation models. Since digital computers constitute the primary physical realizations of numerical computation, and Turing machines define their theoretical limits, the question of Turing computability of PDE solutions arises as a problem of fundamental theoretical significance. The Turing computability of PDE solutions provides a rigorous framework to distinguish equations that are, in principle, algorithmically solvable from those that inherently encode undecidable or non-computable behavior. Once computability is established, complexity theory extends the analysis by quantifying the computational resources required to approximate the corresponding PDE solutions. In this work, we present a novel framework based on least-squares variational formulations and their associated gradient flows to study the computability and complexity of PDE solutions from an optimization perspective. Our approach enables the approximation of PDE solution operators via discrete gradient flows, linking structural properties of the PDE, such as coercivity, ellipticity, and convexity, to the inherent complexity of their solutions. This framework characterizes both regimes where PDEs admit effective numerical solvers in polynomial-time and those exhibiting complexity blowup, where the input data possess polynomial-time complexity, yet the solution itself scales super-polynomially.